Saturday, January 6, 2018

Richard Nowakowski's Tutorial: "Introduction to the Theory of Scoring Games"

Richard gave a great tutorial introduction to to scoring games. There are a lot of issues here I wasn't aware of, including really painful Zugzwangs. ("Anti-Switches"?) E.g: {Final Score: -1 | Final Score: +1} How does this happen?

Consider an Independent Set game played on a cube added to another already-finished game with a score of -3. In the IS game, players choose independent vertices and Left scores a point for the total size of the set. Now, if Left is forced to go first on the cube, Right can choose the opposite vertex, ending the game with a score of 2 (and a total score of -1). If Right goes first, Left can choose another vertex that shares a face with the first. This will force the score to be 4, for a total of +1.

Richard went on to describe many other properties that scoring games can have. He also polled us about whether we should even allow some types of outcomes and games.

Koki Suetsugu: "Normal and Misere play of Multiplayer Games with Preference

Koki looked at what happens in more-than-two-player games when the players have a preference on who they would like to win. He described a result using generalized Nim Sums that I was not aware of. Koki has extended this to misere play by shifting the preference lists any number of spaces to mimic a misere situation. With this, he uses the generalized Nim Sum to solve the game.

Paul Dorbec: "Quantum Nim"

Paul looked at what happens when instead of making a single move in Nim, you make a superposition of two different moves. He points out that it's important to specify the number of sticks you're taking--not the number you're leaving--in each of the submoves. Since some moves may be no longer legal on some parts of the superposition, there are five different versions of the ruleset concerning the allowance of classical moves. (E.g. unsupervised moves are allowed only if they're valid in all of the parts of the current superposition, etc.)

Melissa is really pushing at the border of CGT and Classical (Economic) Game Theory (EGT?). She wants to let both players take their turn simultaneously and wrap new theory around it. For her examples, she used Hackenbush where the resulting move is the intersection of the two chosen options She then defines the outcome classes of terminal positions based on which player has moves left:

Draw: no moves for either player

Positive: Left still has a move

Negative: Right still has a move

Perhaps unsurprisingly, many of the ideas from Classical GT arise as a result of Melissa's ideas, including matrices, expected values, and the benefits of mixed strategies.

Adam Atkinson: "Medieval French Poetry"

This is the only talk that had multiple people climbing to the front of the room to get a better look at all the poems that Adam brought with him. Adam claimed to have brought some old poems that had been rediscovered in 1950, but that was a bit of a ruse. I don't want to fully explain what happened here because I don't want to blow his cover on the chance he's giving a similar talk in any other venue. Nevertheless, maybe we could write some poetry about combinatorial games... but not in English.

Wai Lim (William) Fong: "The Edge-Coloring Game on Trees Played with One More Color"

As I learned at this workshop, there are some really weird and interesting results in the world of Maker-Breaker games. For example, in the edge-coloring game, adding more colors doesn't necessarily help Alice win on graphs. Specifically, exactly one more than the minimum needed to ensure Alice has a winning strategy can actually cause her to lose! William worked on the tree case and showed that if the max degree of the tree is 4, and Alice can win with 4 colors, then she can also win with 5. (6+ colors is already known to be a win for her.) William shows this by constructing Alice's winning strategy!

Urban Larsson: "Playful Game Comparison and Absolute CGT"

Urban and team are looking at other extensions using Absolute CGT. Here he uses an abelian group, called "adorns", of values to assign to terminal positions. This really opens the doors to many rulesets outside of the Winning Ways spectrum. He explains that universes of games are absolute if they are dense and parental. "Normal play is more absolute," he explained. In any absolute universe, comparisons work exactly as you'd want.

Khaled Mama: "Computing the Shapely Values of Graph Games with Restricted Coalitions"

The Shapely value of a coalition describes how much each actor contributes to the coalition. Khaled & team looked at this in an interesting way: what if some coalitions are impossible due to communication barriers? Given some constraints, Khaled can compute these new values using chains across the lattice of coalition chains. To get working games for this, he's focused on some cool graph games.

I learned a ton as a result of these talks! I met a lot of new people that I hope I get to see again. I'm also very interested in everything I learned about Maker-Breaker games. I did not realize there was so much being done on this. Which Maker-Breaker games should I add first to the ruleset table-page?

Friday, January 5, 2018

Rebecca's excellent talk finally convinced me that I could understand this backwards world. Although there isn't all the nice group structure, she showed how to use dicot and dead-end universes to reclaim some nice operations (e.g. invertibility, comparisons, and reductions). It was especially helpful when she explained that Domineering, Hackenbush, NoGo, Snort, and Col are all dead-ending.

More recently, she's applied Absolute Game Theory to get even deeper with dicots and dead ends, including a version of comparisons ("subordinate") and reversibility through ends.

Fionn McInerney: "Spy Games on Graphs"

Fionn showed a cops and robbers variant knows as the spy game. In this game, first a spy is placed on a vertex, then the guards are placed. The spy then moves up to their speed, s. After, the guards can each move up to one space. (It's okay for the guards and spy to co-locate a vertex.) The spy wins if they can escape from the guards on any turn by being at least d+1 distance from all guards. The guards win if they can always prevent this.

Cool complexity connection: It's NP-hard to determine the minimum number of guards needed for their team to win.

Tomoaki Abuku: "Ryuo Nim: a Variant of Wythoff's Nim"

Tomoaki presented a nice variant of Wythoff's Nom, motivated by Shogi. If we consider the piece to move as Shogi's Ryuo, instead of a Chess Queen, then we get a new game. (Ryuo can move as a Rook (Hisya) or as a King.)

Tomoaki found a bunch of Grundy values for game positions and considered many variants.

Gabrielle Paris: "Pre-Grundy Games"

Gabrielle showed an extension to Grundy's Game: given a heap of n tokens, a move consists of splitting it into two different-sized heaps. Gabrielle's Pre-Grundy games allow multiple cuts on one turn: any number in a given set L. Her team has focused on lots of results here, including:

If L does not contain 1, then the Grundy values are arithmetic-periodic, and

If 1 is in L, L has at least two elements, and everything in L is odd, then the Grundy values are periodic!

Lior Goldberg: "Rulesets for Beatty Games"

Lior investigated a problem from 2010 where rulesets were discovered for any Beatty game given by (alpha, beta) where 1 < alpha < 2 and 1/alpha + 1/beta = 1. The problem is that the rulesets were also very complicated. Lior has found a nice ruleset for Beatty games where alpha < 1.5 and alpha >= 1.5. Unfortunately, there are some examples where the ruleset must explicitly reference alpha. He also showed how to get some specific rulesets that look like t-Wythoff generalizations.

Valentin Gledel: "Maker-Breaker Domination Game"

Valentin introduced a new Maker-Breaker version of a dominating set game. Each player choose an unchosen vertex. The Dominator (player) adds them to a potential dominating set (a set of vertices that contains and are neighbors to all vertices). The Staller (player) chooses nodes to remove as options of the Dominator. The Dominator wins if the final set dominates the entire graph. Valentin and his team showed that there are only three outcome classes (N - Fuzzy, D - Dominator wins, S - Staller wins) and showed what happens for all disjoint unions and joins. He showed that it's easy to decide on cographs and trees, but PSPACE-complete in general.

Clement and team investigated the Maker-Breaker Graph Coloring Game: Alice (Maker) and Bob (Breaker) take turns coloring vertices from one of k colors, following the regular graph coloring rules. If, in the end, all vertices are colored, then Alice wins. Otherwise, Bob wins.

The Nordhaus-Gaddum inequality is about the chromatic number of a graph. Clement applied these to help determine strategies for Alice. Clement used this to create new inequalities describing when Alice can win!

Dominique found a bunch of variants of the Graph-Coloring game by varying:

Who goes first (Alice or Bob)

Whether Alice or Bob (not both) can pass, and

Whether missing a turn is allowed.

Then, for any variant, he and his team define a nice property: a graph is "game-perfect" for one variant exactly if the game-chromatic number equals the size of the largest clique. They have lots of results for the different variants! This talk was the first I've seen that used the word "perfectness". :)

Thursday, October 26, 2017

The Games and Graphs Workshop in Lyon was outstanding! There were so many talks, but I think I understood at least the basics of most of them. The first talks were on Monday.

Aviezri Fraenkel: "Problems and Results in Combinatorial Games and Combinatorics"

Aviezri presented a talk about games with some elements of (Economic) Game Theory (EGT). For example, he considered playing Geography on a binary tree where all terminal nodes are after Alice's turn, except for a great many on the lowest level. At each level, Alice has one instantly-winning move, but what if there are many other moves and Alice can't differentiate between them? Now she is essentially choosing at random because of some hidden information. With enough of these extra edges, she is not going to be able to win the game, despite having a winning option each turn.

Aviezri showed many other connections to EGT and called for future work on infinite games in the partizan, scoring, and misere realms.

Craig Tennenhouse: "Three Graph Games"

Craig presented three new games: Tumbleweeds, Deletion/Contraction, and Total Conversion. Tumbleweeds is a cool variant of Hackenbush without a static root. Instead, whenever a player chooses an edge that breaks the graph up into multiple connected components, that player chooses either of the components to completely discard!

The partizan game Deletion/Contraction is played on a multigraph (but no loops). The Left player moves by deleting an edge on their turn, while the Right player chooses an edge to contract (removing any loops that are created).

In Total Conversion, the position is a graph with a non-zero game embedded into each edge and each vertex colored either Red or Blue. To take a turn, the player must make a move on each game where the edge has ends colored in opposing colors. Edges with exactly 0 are then removed and both vertices are subsequently repainted with that player's color.

Craig already drew attention with these games and, as always, I suspect he has about three new projects as a result of the conference.

Loic Cellier: "The Game of Hex"

It always seems like there are new things to learn about Hex. Loic presented just about everything I knew and lots that I didn't. He covered the history---I did not know that Einstein was a supporter of the game!---the connection to Y, and even presented a variant of the Pie Rule, the Swap Rule: Here, instead of swapping the identity of the players (if Player 2 wants), instead just swap the color of the piece on the board. This is, of course, not always equivalent; it prevents the first player from playing in a position that would be too good for their opponent as well.

I also (finally) learned why a Y game must end, by the self-reduction of each vertex into a hex on a slightly-smaller board. This is surprisingly elegant and easy to explain, but with that little twist that makes it amazing. It may be more likely to be in The Book than the Hex Theorem (that no Hex game can end in a tie.

Milos Stojakovic: "Maker-Breaker Games Played on Random Graphs"

Milos talked about Positional Games: maker-breaker games on a finite set of elements, X, where F, the winning set, is a subset of the power set of X. The players alternate turns claiming one element of X. Maker then wins if they claim all the elements in some set in F; Breaker wins otherwise. Milos paid special attention games where X is the set of edges in a complete graph.

Unfortunately, for many common sets F (e.g. set of Triangles) it is too easy for Maker to win. To balance things out, he considers adding biases for each player to change the frequency of turns. (E.g. Maker takes 3 turns, then Breaker takes 7.) The idea he took off with was to begin the game by removing a number of random edges. If each edge remains with probability p, how low does p have to be before Breaker can (usually) win? He calls this the Threshold Probability, and looked in to determining this value.

Mirjana Mikalacki: "Fast Winning in Positional Games on Graphs"

Mirjana is also interested in positional games, but is looking for other ways to give Breaker a chance to win. Instead of removing some edges, she considers a "fast" variant where Maker only gets to make a certain number of moves before the game ends in Breaker's favor. (Breaker still gets to move as normal too.)

Mirjana also combined this with the biases mentioned above. These "combination" fast games generalize positional games significantly.

Sunday, July 30, 2017

Elwyn Berlekamp's video-talk at Fundy and Games, "A Program of Introductory Videos on Combinatorial Games", alerted me to something I didn't realize: Elwyn's been creating a bunch of videos to introduce CGT. Since the meta-video presentation, I've been watching the videos.

He's using some good philosophy towards introducing concepts:

Examples come before the abstractions. It's easier to motivate the abstract notions once you've seen some concrete examples.

Delay notation as long as possible. The non-standard notation used while still showing the initial concepts works just fine. For example, instead of using N and P (outcome classes), the impartial videos start off with just checkmarks (for N-positions) and circles (P).

Here's the sequence of videos on impartial games. They use an impartial version of chess where each piece occupies it's own board and each piece can only move towards the southwest corner. The first video describes what happens with one King, and the following videos show what happens as other pieces get added, each time adding a new concept to the theory of games.

Friday, July 28, 2017

For the tournament game at Fundy and Games, we played a variant of Domineering known as NoCanDo. (We spent the first day deciding on the name.) This is a very nice mix of Domineering with a little splash of NoGo: each domino on the board must be adjacent to at least one uncovered square. That's it. You're not allowed to play a domino if it would be completely surrounded or if would cause another domino (already played) to be completely surrounded.

This little rules change makes a huge difference!

You can place pieces aggressively to force an opponent to leave spots open.

You need three spaces in a line for a free move, instead of two. (And sometimes you can't play in those three anyways because it blocks other dominoes.) Five in a line can often net you two plays, but I don't think I ever had a group worth three.

The "liberty" rule is still different from Go and NoGo, because each domino needs the free space, not just each connected group of similarly-oriented dominoes.

When the game breaks up into subregions, sometimes they're not all independent.

We held human and computer tournaments at Fundy and Games. RJN won the human tournament, and my program won the computer tournament. Richard handily beat my computer player to win the overall title.

Thursday, July 27, 2017

One of the games Ludus has used in their national tournaments is Slimetrail. This game is simple enough that it can be played by a wide age range, but there are many complicated strategies that can be used by more advanced players.

Slimetrail is played on a connected graph, with one vertex colored Blue, another colored Red, and a third vertex with a moveable piece or token which will create the trail of slime. The two players alternate turns moving the token one space, then marking the previous space (where the token was) a third color (usually green). The token can never be moved back to one of these "slimed" spaces.

A player wins when the token is moved onto the space of their color. Since we want to make sure that one player can still win, it's not allowed to move the token to a space where it can't reach at least one of the two goals.

In all the examples I've seen, Slimetrail is played on a grid, with adjacencies in all 8 directions. Apparently, it's also played on hex grids.

I wrote a playable version using Javascript. My auto-AI players are terrible at this game, but you can still try it out. In order to make this strictly-combinatorial (the last player wins) I altered the rules slightly so that you can't move to your opponent's goal space.

(Edit: the link to the game didn't auto-clickableify, so I added a clickable link.)

Combinatorial Game Man

artist: Molly Dannaher

What is This?

This blog is devoted to Combinatorial Game Theory! Combinatorial games are two-player games with no randomness, perfect information (no one has any hidden information) and no draws allowed, though sometimes topics stray into other types of games as well.

Please let me know if you are interested in either writing a guest post or suggesting a topic. I would like this to be more reflective of CGT as a whole and not restrict it to my take on things. Additionally, please help me fill out the table of game properties, either by suggesting games or letting me know about results!